A multiple x-ray-source array (MXA) system with a planar two-dimensional source distribution for digital breast tomosynthesis

Abstract

Background:

Digital breast tomosynthesis (DBT) has outpaced digital mammography in clinical adoption in the United States; however, substantial technological limitations remain to image quality in DBT, including undersampling from a one-dimensional (1D) scan geometry, x-ray source motion during acquisition, and patient motion artifacts from long exam times.

Purpose:

A thermionic cathode x-ray system employing two-dimensional (2D, planar) multiple x-ray-source arrays (MXA) is proposed to improve DBT image quality.

Methods:

A 1D MXA, consisting of a linear array of thermionic cathodes was used to simulate a 2D MXA. The 1D MXA included 11 focal spots separated by a distance of $\Delta d=23\text{mm}$. The 11 cathodes were paired with 11 molybdenum 50 mm diameter anode disks, mounted on a rotating shaft within a single vacuum enclosure. Image quality was investigated as a function of MXA configuration by integrating the 1D MXA with a 200 × 250 mm² flat panel detector at a source-to-detector distance of 630 mm, resulting in a 20° tomographic arc. To simulate a 2D MXA, the detector (with phantom) was translated orthogonally to the linear array by a distance $\left(\delta \right)$ ranging from $\delta =0\text{mm}$ (conventional 1D) to $\delta =57\text{mm}$. All sources operated at 30 kV with 80 mA and 4.5 mAs/pulse, yielding ~100 mAs per DBT dataset. DBT reconstructions involved 22 projections and used filtered backprojection with a ramp and Hann apodization filter. Volumetric reconstructions for each source were weighted by sampling differences between sources, and averaged. Image quality was assessed in terms of contrast-to-noise ratio (CNR), background clutter noise and power spectrum, and slice sensitivity profile (SSP) using a set of physical phantoms, including: (i) contrast-detail signals coupled to spherical clutter (PMMA in air); (ii) an SSP phantom; (iii) a commercial “breast” phantom (CIRS BR3D, Sun Nuclear, Norfolk, VA); and (iv) bovine muscle.

Results:

Background clutter noise amplitude reduced monotonically from the 1D MXA (${\sigma }_{\text{clutter}}=5.9$ A.U., $\delta =0\text{mm}$) and 2D MXA arrays with increasing $\delta$, with statistical significance between the 1D MXA and 2D MXA with $\delta =57\text{mm}$ (${\sigma }_{\text{clutter}}=5.0$ A.U., $p<0.001$). The contrast-detail/clutter phantom demonstrated CNR from the 2D MXA $(\delta =57\text{mm})$ outperforming the 1D MXA in all combinations of contrast and detail.2D power spectrum analysis of clutter demonstrated a pronounced Fourier domain null cone for the 1D MXA in the anterior field-of-view (away from the 1D MXA position), whereas the 2D MXA geometry $(\delta =57\text{mm})$ did not exhibit the null cone. The SSP was 15%−50% narrower (FWHM) for the 2D versus the 1D geometry, across all reconstruction setups.

Conclusions:

The advantages of a 2D source geometry for DBT imaging were demonstrated quantitatively compared to a conventional 1D line of x-ray sources. The improvement in the 2D geometry was attributed both to improved Fourier domain sampling and reduced SSP. We conclude that 2D MXA sources have the potential to substantially improve DBT imaging in comparison to existing commercial DBT systems.

1 |. INTRODUCTION

Since the implementation of tomosynthesis in the 1930s, it was known that relative to 1D motion, 2D motion of the x-ray source resulted in better blurring of the shadows of structures both above and below the focal plane. This significantly improved image quality by increasing the tomographic resolution and, depending upon scan angle geometries, better constraining the slice sensitivity profile (SSP).^1,2 These improvements in image quality resulted in the development of commercial systems with exotic 2D motions such as “trispiral” (Stratomatic tomography system—CGR Medical Corporation/CGR-Thomson) and “hypercycloidal” (Polytome tomography system—Philips). More recent studies demonstrated the improvement in image quality as the result of circular source motion, relative to the 1D (i.e., arc) motion used in current commercial digital breast tomosynthesis (DBT) systems.^3–5

Such understanding born from classic motion tomosynthesis suggests a means for improved performance in the modern context of DBT via source-detector sampling geometries beyond that of a single line or arc. Despite the potential benefits to DBT image quality provided by 2D source acquisition geometries, their adoption is hampered in part by the mechanical complexity associated with introducing a second degree of freedom to the positioning system of a single x-ray source and the associated increase in manufacturing and maintenance costs. More importantly, implementing 2D mechanical motion would likely increase scan times significantly, which are already long with current DBT systems. The shortest scan time of FDA-cleared commercial systems is nearly 4 s (Hologic), with others ranging from 7 to 25 s.⁶ Detailed analysis has shown that to minimize patient-motion artifacts, DBT scan times should be < 2 s⁷ (i.e., similar to requirements for conventional digital mammography). Artifacts due to patient motion are a key reason scan time is considered to be an important parameter characterizing a DBT system.^8,9 Additionally, DBT systems featuring shorter scan times (e.g., Hologic) use acquisition protocols with continuous motion of the x-ray source, avoiding longer acquisition times associated with step-and-shoot techniques, but inducing focal spot blurring,⁶ which reduces spatial resolution. The potential for improvement of DBT image quality with stationary x-ray multi-source arrangements is therefore threefold: (i) enabling sub-second image acquisition which reduces patient motion, (ii) eliminating source motion during x-ray pulsing which occurs with mechanically scanned DBT systems—improving spatial resolution, and (iii) improving volumetric sampling by distributing the x-ray elements comprising the stationary source in 2D spatial patterns.

For nearly 20 years the development of stationary sources for DBT has focused on using field emission cathodes^10,11 such as carbon nanotubes (CNTs). Although there has been recent interest¹² in addressing lifetime limitations that have hindered the incorporation of these cathodes in clinical systems,¹³ the improvements have been made with stationary anode x-ray tubes. The use of stationary anodes limits the source’s maximum x-ray flux, leading to clinically unacceptable long exposure times, particularly for low tube voltage (~28–32 kV) applications such as DBT. Limitations from stationary anodes could be mitigated via integration of rotating anode assemblies with CNT cathode designs, however, such integration is hampered by the need for the ultrahigh vacuum environments necessary to yield longer CNT cathode lifetimes. Compared to stationary anodes, rotating anode designs limit the bake-out temperatures of the tube bearings and have inherent long-term anode outgassing rates during operation,¹⁴ which limit the achievable vacuum level. On the other hand, thermionic x-ray cathodes do not require an ultrahigh vacuum environment and are thus more suitable for integration into rotating anode designs.

Recent work has demonstrated the feasibility of building reliable, high-output x-ray sources in the form of arrays of discrete, individually addressable thermionic-cathode-based x-ray source elements contained within a single vacuum envelope,¹⁵ referred to as a multiple x-ray-source array (MXA). The MXA employs tungsten filament-type cathodes for reliability and long lifetime, focus-cup-biased control for modulation of the x-ray emission from the individual source elements, and a rotating anode for increased x-ray output over that achieved by stationary anodes. Figure 1a illustrates the MXA system design for a conventional (i.e., 1D) DBT system using an array of 11 individually addressable source elements. For DBT, the 1D MXA provides a stationary x-ray source, and relative to existing FDA-approved systems, it poses a variety of potential advantages, including: (i) elimination of the mechanical motion required for DBT imaging, and (ii) decreased image acquisition times (within detector frame rate limitations), leading to reduced patient-induced motion artifacts. Furthermore, 2D DBT scans, with scan times of <1 s, could potentially be implemented by fitting the 1D MXA with a second array of cathodes, sharing the same set of anode discs, as shown in orange in Figure 1a.

FIGURE 1.

(a) The 1D MXA is an array of grid-controlled thermionic cathodes (cathode array 1) opposite a corresponding array of anode disks on a common rotating shaft. Each of the eleven x-ray source elements in the linear array can be individually addressed. The 1D MXA can be extended to a 2D MXA by the addition of a second set of grid-controlled cathodes opposite the array of anode disks (labeled “cathode array 2″, in orange). (b) The imaging geometry of the 2D MXA is shown in comparison to a conventional 1D DBT configuration by depicting the relative position of the x-ray focal spots (red) when viewed from behind the source looking toward the breast as projected on the detector.

The resulting 2D MXA provides 2D DBT sampling with two linear source geometries, one located close to the chest wall (posteriorly) and one towards the nipple (anteriorly). Thus, the conventional scan path of the x-ray tube (as implemented in FDA-approved DBT systems) creates a DBT acquisition in only one dimension when viewed from above whereas the 2D MXA creates a planar, rectangular sampling pattern of the breast for DBT, as illustrated in Figure 1b. The maximum sampling density for DBT, and the minimum separation between the linear arrays for 2D sampling, are set by the distance between the individual anode disks $\left(\Delta \text{d}\right)$ and the diameter of the anode disks $\left(\delta \right)$, respectively. The shared anode geometry results in a compact embodiment and minimal overhead in hardware and construction costs. Compared to the combination of two individual MXAs, the proposed dual-cathode concept results in a single rotating shaft, shared anode material, and a single vacuum enclosure. The shared anode concept is limited to moderate values of $\delta$, with large $\delta$ yielding bulky setups that would be more difficult to integrate into compact DBT systems. Preliminary studies have shown that systems with $\delta$ up to ~120 mm are practically feasible. Although having two cathodes adds extra heat to the anode, the tube loading conditions in DBT are modest. Additionally, the common shaft arrangement enables the distribution of heat across 11 different anodes. Furthermore, the two electron pulses received by each anode are separated in time, reducing heat dissipation requirements.

The work reported here involves experiments designed to quantify the impact of the geometry of the planar, 2D MXA source distribution on DBT image quality, and to investigate the initial optimization of the design of the 2D MXA.

2 |. METHODS AND MATERIALS

2.1 |. Source geometry and acquisition arrangement

The MXA source used in these studies was a linear array of eleven grid-bias-controlled thermionic cathodes operating against a single rotating shaft with 11 anodes. Each source element in the array defines a focal spot with an approximate size of 0.3 mm, an anode angle of 12 deg, and a separation with adjacent elements of $\Delta d=23\text{mm}$. The source design and operating parameters of the 1D MXA have previously been described in detail.¹⁵ The 1D MXA was integrated into the experimental imaging arrangement illustrated in Figure 2, which includes a PaxScan 2520 flat-panel detector (Varex Imaging Corp., Salt Lake City, UT) with a 1536 × 1920 matrix size and a pixel size of 0.127 mm × 0.127 mm. While this detector is perhaps not optimal for clinical DBT applications, it is useful in demonstrating the proof-of-principle for purposes of the current work. The flat-panel detector was mounted on a precision linear translation stage to enable the investigation of several 2D MXA configurations using detector (and phantom) translation relative to the 1D MXA source.

FIGURE 2.

Photograph of the 11-source 1D MXA DBT system used to produce 2D MXA DBT scans.

The geometrical configuration of the experimental setup, shown in Figure 3a, was designed to emulate clinical DBT scanners. The source-to-detector distance (SDD) was set to 630 mm, defining a total scan angle of 20° for the 11-element stationary 1D MXA with a source separation of $\Delta d=23\text{mm}$. The geometrical configuration of the experimental 1D MXA DBT system yields a number of projection views and a scan angle between that of the GE Senoclair (9 projections over 25°) and the Hologic Dimensions 3D (15 projections over 15°).⁶ As illustrated in Figure 3a, 2D MXA configurations were emulated by translating the detector and object to be imaged from their initial position in steps of 19 mm to simulate three distinct 2D MXA configurations with a distance between arrays $\left(\delta \right)$ of 19, 38, and 57 mm (see Figure 3b). As illustrated in Figure 1a, $\delta$ is coincident with the diameter of the disc anodes in the proposed configuration. Note that when comparing equidose configurations, adding anterior source positions reduces the dose delivered to a small volume near the chest wall, as shown schematically in Figure 3a.

FIGURE 3.

Schematic of 2D MXA imaging geometry. (a) Side-views of the MXA-Phantom-Detector arrangement showing how translation of the detector relative to the MXA was used to simulate 2D DBT scans. The beam shaped by the posterior MXA is depicted in dark blue, the anterior MXA beam is shown in light blue, and the volume of full coverage by the 2D MXA is marked by the black contour. (b) View of the detector from the (central) MXA source showing the 44 unique source positions above the detector. The sixth of the eleven x-ray source elements was centered on the detector. (c) Photograph of the phantom used to calibrate the geometry of the 2D MXA DBT system. The 6.4-mm-thick polymethylmethacrylate (PMMA) sheets are 18 cm on each side. The phantom contains a structured set of 3-mm ball bearings (BB) and washers with 3 mm inner diameter (ID) and 6 mm outer diameter (OD).

Image acquisition was performed with a 30 kV spectrum and anode current of 80 mA with a pulse length of 56 ms, for a total of 4.5 mAs/projection. No extra beam filtration was added to the MXA output window (1.0 mm Al). This x-ray beam was modeled and found to deliver an estimated 18.0 keV effective energy with an HVL of 0.56 mm Al,¹⁶ very similar to the beam quality measured on several DBT systems at UC Davis.

A total of 4 datasets were acquired per experiment, one with a single 1D MXA configuration, equivalent to conventional (linear) DBT acquisitions, and three with 2D-MXA acquisitions, for the three values of $\delta$ listed above. To maintain similar radiation dose conditions across acquisition protocols, for the single 1D MXA configuration $(\delta =0\text{mm})$, two independent projection views were acquired per MXA element, and the 2 images were included independently in the reconstruction. Therefore, all protocols involved the acquisition of a total of 22 projection images, resulting in 100 mAs/scan. A reference single-position mammography view was acquired for baseline comparison to all DBT configurations. Acquisition for the mammography geometry used 22 x-ray pulses from the single central x-ray focal spot, amounting to an equivalent total of 100 mAs. Individual frames were subsequently averaged to generate the final mammographic view.

2.2 |. System calibration and image reconstruction

Calibration of the geometrical parameters of the imaging system for reconstruction of DBT volumetric images was achieved with a custom calibration phantom. As shown in Figure 3c, the phantom consisted of a square grid of 3 mm diameter steel spheres regularly spaced in 25.4 mm increments in both directions (approximately aligned with the vertical $\left(y\right)$ and horizontal $\left(x\right)$ directions of the detector) and adhered to a 6.4-mm-thick sheet of polymethylmethacrylate (PMMA). The PMMA sheet holding the grid of spheres was placed above a second grid of annular steel washers having a 3.0 mm inner diameter and 6.0 mm outer diameter. The centers of the washers were regularly spaced at 25.4 mm increments in both directions and adhered to a 6.4-mm-thick PMMA sheet. The two grid patterns were separated by a 25.4 mm air gap, and the centers of the steel spheres and washers were aligned one above the other.

Geometrical calibration was performed using a registration-based approach for numerical optimization of the relative position between each MXA element, the imaged volume, and the stationary detector. The geometry for each source element-detector pair was described as a projection matrix with 9 degrees-of-freedom (DoF), accounting for the position of the source element (3 DoF), and the position of the detector (6 DoF), relative to the center of the calibration phantom, which is used as the origin of coordinates, and considered fixed across views.

A CAD model of the sphere-grid arrangement on the geometry calibration phantom was built and converted into a high-resolution binary volume ${(\mu }_{\mathit{\text{cal}}})$ of size 4096 × 4096 × 640 voxels and an isotropic voxel size of 0.0625 mm. The resulting volume was used for computation of synthetic projection views as a function of the set of projection matrices (${\Pi }_{\text{i}}$, with $i=\left[1,\dots ,N_{\mathit{\text{MXA}}}\right]$), where $N_{\mathit{\text{MXA}}}$ is the number of sources. Simulated projections (digitally reconstructed radiographs) were computed for the MXA configuration as ${\hat{y}}_i\left({\Pi }_i,{\mu }_{\mathit{\text{cal}}}\right)=\mathit{A}\left({\Pi }_i\right){\mu }_{\mathit{\text{cal}}}$, with $\mathit{A}$ denoting a forward projection operator, and geometric calibration was obtained by iterative optimization of the image similarity metric computed between the real images $\left(y\right)$ and the simulated projection views $\left(\hat{y}\right)$, giving projection matrices:

$$\Pi =\underset{\Pi }{\text{arg}\text{max}}\sum _{i=1}^{N_{\mathit{\text{MXA}}}}\mathit{\text{GO}}\left[y_i,{\widehat{y}}_i\left({\Pi }_i,{\mu }_{\mathit{\text{cal}}}\right)\right]$$ (1)

where $\mathit{\text{GO}}$ is the gradient orientation similarity metric, which has been previously applied to a wide range of 3D/2D registration problems.^17,18

Image reconstruction for the calibrated 1D and 2D MXA configurations was performed in volumes of 2048 × 2048 × 256 voxels (0.1 × 0.1 × 0.5 mm³ voxel size) with filtered backprojection (FBP), assuming a linear cone-beam geometry per individual 2D MXA position. Following previous literature for FBP-based reconstruction in DBT,^19,20 we assumed that each 1D MXA configuration defines a wedge-shaped sampling region in the frequency space of the central tomosynthesis slice (i.e., the plane orthogonal to the detector and including the 1D MXA). According to this model, individual projection views were filtered with a modified ramp filter acting along the detector rows and given by $H_{\mathit{\text{RA}}}\left(f_x,f_z\right)=2\cdot \text{tan}\left(\theta \right)\sqrt{f_x^2+f_z^2}$, with $\theta$ describing the angular range covered by the MXA (20° in this work), and $f_x$ and $f_z$ denoting the frequency in the lateral and superior-inferior directions of the reconstructed volume (denoted $x$ and $z$ in Figure 3, respectively). To reduce the amplification of noise induced by the ramp filter, a Hann apodization window was applied on the lateral frequency axis, $H_{\mathit{\text{AP}}}\left(f_x\right)=0.5\cdot \left[1+\mathit{\text{cos}}\left(\pi f_x/f_{\mathit{\text{AP}}}\right)\right]$, with apodization cutoff frequency, $f_{\mathit{\text{AP}}}=0.8\cdot f_{\mathit{\text{xN}}}$, where $f_{xN}\left(=5{\text{mm}}^{-1}\right)$ is the Nyquist frequency in the lateral direction.

Finally, to reduce interslice clutter, a slice thickness filter was applied before backprojection. The slice thickness filter was also modeled following a Hann window, as $H_{\mathit{\text{ST}}}\left(f_z\right)=0.5\cdot \left[1+\mathit{\text{cos}}\left(\pi f_z/f_{\mathit{\text{ST}}}\right)\right]$, where $f_{\mathit{\text{ST}}}$ is the interslice cutoff frequency. Following previous literature,¹⁹ the cutoff frequency of the slice thickness filter was set as a fraction of the Nyquist frequency in the superior-inferior axis, $f_{zN}\left(=1{\text{mm}}^{-1}\right)$, following $f_{\mathit{\text{ST}}}=\beta \cdot f_{zN}$, with $\beta$ ranging between 0.1 and 0.9, with low values imparting a larger reduction in high-frequency components associated with interslice contamination. The reduction in interslice contamination was, however, coupled to an overall loss in spatial resolution. The nominal reconstruction setup involved $\beta =0.3$, used for all experiments for which explicit values of $\beta$ are not listed.

Individual MXA reconstructions were averaged in the volumetric domain with voxel-wise weighting to account for biases in reconstructed attenuation values arising from non-homogeneous sampling and different cone-angles for voxels sampled by more than one source. The weighting volume, $\mathit{\kappa }$, was computed in a manner similar to previous approaches to multi-source cone beam CT²¹ according to the following expression:

$$\mathit{\kappa }=\left(\sum _{\text{j}=1}^{N_{\mathit{\text{MXA}}}}{\mathit{A}}_j^{T}F{T}^{-1}\left(H_{\mathit{\text{RA}}}\cdot H_{\mathit{\text{AP}}}\cdot H_{\mathit{\text{ST}}}\cdot \mathit{\text{FT}}({\mathit{A}}_j\left(\mathbf{1}\right)\right)\right)$$ (2)

where $\mathit{A}$ and ${\mathit{A}}^T$ are the forward and backward projection operators, $\mathit{\text{FT}}$ and $F{T}^{-1}$ denote the direct and inverse one-dimensional Fourier transforms taken across the rows of the detector, and $1$ is a volume of all ones with the same size as the reconstruction volume.

2.3 |. Experimental assessment

Four phantoms were used for assessment of the impact of 1D- and 2D-MXA configurations on image quality: (1) A hole-stimulus phantom including anatomical clutter, for quantitative experimental studies of noise, contrast, interslice clutter, and background power spectrum; (2) an SSP phantom for quantification of interslice spatial resolution; (3) an anatomically realistic breast phantom allowing qualitative evaluation of image quality for a variety of imaging tasks; and, (4) a biological phantom, made of bovine muscle tissue with interstitial fat.

2.3.1 |. Hole-stimulus phantom

The hole stimulus phantom is illustrated in Figure 4a. The phantom consisted of a 6.40-mm-thick sheet of PMMA featuring a set of six linear arrays of four flat-bottomed holes (denoted ‘stimuli’ below) equally spaced 19 mm between consecutive centers and running parallel to the MXA direction (i.e., along the $x$-axis as shown in Figures 2 and 3b). The four stimuli diameters were: 6.35, 4.76, 3.18, and 1.59 mm. The largest stimulus was located at the right-most position in the array for odd-numbered array positions (1,3,5—see Figure 4a), and at the left-most position for even-numbered array positions (2,4,6—see Figure 4a). The arrays were numbered from posterior to anterior positions, denoted ‘proximal’ $(y=0)$ and ‘distal’ $(y>0)$ in the discussion below to denote their relative location from the position of the central ray for a conventional 1D configuration.

FIGURE 4.

(a) Photograph of the hole-stimulus phantom consisting of flat-bottom holes having various diameters and depths in a 6.4-mm-thick sheet of PMMA, 16.5 cm on a side. (b) Photograph of the clutter phantom consisting of a PMMA box (16 cm on a side with an interior thickness dimension of 1.9 cm) holding a mix of PMMA spheres of three different diameters [4.8, 6.4, and 9.5 mm]. (c) Anatomically realistic phantom (CIRS Breast phantom) with high-contrast, high-frequency CaCO₃ patterns, a surrogate for microcalcifications, and soft-tissue masses of increasing diameter.

To provide a variation in image contrast as a function of the location of the array in the DBT field of view, the depth $\left(h\right)$ of the holes (“stimuli”) was dependent on the array position. Three depth values were used: high-contrast $(h=6.40\text{mm})$, medium-contrast $(h=3.18\text{mm})$, and low-contrast $(h=1.59\text{mm})$. High-contrast stimuli were used in arrays 1 and 4, medium-contrast stimuli were used in arrays 2 and 5, and low-contrast stimuli were used in arrays 3 and 6 (see Figure 4a).

2.3.2 |. Clutter phantom

To simulate the impact of anatomical clutter on image quality, a pair of clutter phantoms was used in conjunction with the hole stimulus phantom. The clutter phantom design, illustrated in Figure 4b, followed previous approaches for the evaluation of the effect of anatomical clutter on tomosynthesis image quality.^22,23 Each of the two clutter phantoms consisted of a PMMA box with 6.4-mm-thick walls containing a random assortment of PMMA spheres with a density of 1.18 g/cm³, and with three different diameters (denoted $D$): 4.8, 6.4, and 9.5 mm. The thickness of the cavity filled with PMMA spheres was 19 mm. For image acquisition, the hole stimuli phantom was inserted between two clutter phantoms. To maximize the impact of anatomical clutter, the interspace material in the spherical clutter phantom was filled with air.

Quantitative evaluation of interslice clutter was obtained using metrics of anatomical clutter noise for the hole-stimulus and clutter phantom, as a function of the 2D MXA spacing $\left(\delta \right)$. Anatomical clutter noise was measured in the tomosynthetic image plane containing the stimuli. Under the assumption that the anatomical clutter noise dominated over quantum noise for the experimental setup, anatomical clutter noise was measured as the standard deviation of voxel values inside local regions of interest (ROIs). 48 ROIs with a size of 8 × 8 mm² (80 × 80 voxels), arranged on a rectilinear grid of 8 × 6 ROIs, placed across the complete field-of-view in the DBT plane were used for estimating clutter noise. The statistical significance of differences in noise clutter across the different MXA configurations was evaluated with the Wilcoxon rank-sum test, with $p<0.05$ as the significance level.

Conspicuity of the image stimuli in the presence of interplane clutter was quantitatively assessed using the local contrast-to-noise ratio (CNR). The contrast was estimated as the absolute difference between the average voxel value inside the stimuli and the average signal value across the background ROIs used for the estimation of anatomical clutter. The average signal in each stimulus was obtained using a circular ROI centered at the central voxel of the stimulus with a diameter equal to 90% of the diameter of the stimulus of interest. Noise was estimated as the average standard deviation across all background ROIs.

The impact of 2D MXA configurations on the local frequency content of the reconstructed DBT planes was evaluated using measurements of background power spectra $\left(S_B\right)$ within a reconstructed plane focused on the clutter phantom. Two-dimensional slice power spectra were measured, one for locations close to the chest wall $(y\approx 0)$ and a second at anterior locations $(y\gg 0)$. Power spectra were obtained using an approach similar to Gang et al.,²³ on a set of partially overlapping ROIs of 24 mm × 24 mm (240 × 240 pixels), arranged on a regular grid of 12 × 3 ROIs, with 66% overlap. The size of the ROIs was selected to cover a region large enough to capture the clutter distribution from the heterogeneous clutter features, but small enough to reduce the impact of low-frequency signal variations. To further mitigate low-frequency image components, each ROI was detrended by fitting a 2D, second-order polynomial. The background power spectrum was obtained as the ensemble average of the square magnitude of the 2D Fourier transform (denoted $\mathit{\text{FT}}$), computed across ROIs as:

$$S_B\left(f_x,f_y\right)=\frac{1}{{\overline{\mathit{\text{ROI}}}}^2}\frac{{\Delta }_x{\Delta }_y}{N_x{N}_y}\frac{1}{{\mathit{\text{Norm}}}_W}$$

$$\langle {\left|\mathit{\text{FT}}\left[W\left(x,y\right)\cdot \left(\mathit{\text{ROI}}\left(x,y\right)-\overline{\mathit{\text{ROI}}}\right)\right]\right|}^2\rangle$$ (3)

where $\overline{\mathit{\text{ROI}}}$ is the ensemble average across ROIs in the estimation to normalize signal power across configurations, and ${\Delta }_x\left({\Delta }_y\right)$ and $N_x\left(N_y\right)$ are the voxel size and ROI extent, respectively, in the $x\left(y\right)$ directions. $W(x,y)$ is a Hann tapering window applied on the individual ROIs to mitigate the effect of spectral leakage, given by:

$$W\left(x,y\right)=\{\begin{array}{l}0.5\cdot \left[1+\mathit{\text{cos}}\left(\frac{\pi r}{N_x/2}\right)\right]\text{for}r\le N_x\hfill \\ 0\mathit{\text{otherwise}}\hfill \end{array}$$ (4)

where $r=\sqrt{x^2+y^2}$ is the distance of the specific pixel from the center of the ROI, measured in pixels. The power spectra were normalized by the energy of the tapering window, computed as the average of $\left|W^2\right|$ across the ROI, reflected in ${\mathit{\text{Norm}}}_w$. Note that the 2D slice power spectrum, computed according to Equation (3), corresponds to a DBT “slice extraction” operation from the full 3D volume, akin to common visualization procedures for DBT, in which the radiologist browses slice-by-slice across the volume. The power spectrum $\left(S_B\right)$ computed following the slice extraction operation therefore differs from the calculation of the full 3D power spectrum.^23–25

2.3.3 |. Slice sensitivity profile

Quantitative assessment of DBT interslice contamination for high-frequency imaging tasks in the presence of anatomical clutter was evaluated using the SSP computed on a 1 × 1 mm² brass sheet of 25.4 μm thickness, placed approximately at the center of the system field of view. The interslice resolution was quantified as the average and standard deviation of the full-width at half maximum (FWHM) of the SSP, computed for six consecutive coronal slices of the brass sheet. The statistical significance of differences in the FWHM values of the SSP across the different MXA configurations was evaluated with the Wilcoxon rank-sum test, with $p<0.05$ as the significance level.

2.3.4 |. Anatomically realistic phantoms

Validation in more anatomically motivated samples was achieved using a commercial quality-assurance phantom (CIRS BR3D breast imaging phantom Sun Nuclear, Norfolk, VA), specifically designed for evaluation of image quality in mammography and DBT. The phantom, illustrated in Figure 4c, consists of a set of 180 × 100 mm semi-circular-shaped slabs approximating breast anatomical clutter with a swirl pattern of tissue equivalent materials mimicking 100% adipose and 100% glandular tissues. The swirl pattern in the slab was designed to reproduce anatomical patterns with approximately 50/50 ratio by weight between adipose and glandular tissues. The phantom included a set of six arrangements each containing six impulse-like stimuli, acting as surrogates for microcalcifications, made of CaCO₃. The diameter of the stimuli ranged from a minimum of 0.13 mm to a maximum of 0.40 mm.

Microcalcification arrangements were placed in the flat region of the semicircular slab, assumed to be posterior in the breast. A second set of six features simulated solid, spheroidal, low-contrast masses, surrogates for breast carcinoma. The spheroids ranged from 1.80 to 6.30 mm in diameter. The mass-like features were located anteriorly in the semicircular slab. Finally, a set of six angled linear fibrils with a length of 10 mm and increasing thickness, ranging from 0.15 to 0.60 mm, was placed in the central locations of the slab, allowing evaluation of the visibility of linear anatomical features. For imaging, the phantom was arranged as a stack containing 5 clutter slabs, in combination with the image quality slab, resulting in a total thickness of 60 mm. The image quality slab was placed in the fourth position in the stack, that is, [source ↔ ●●●○●● ↔ detector] where O is the image quality slab. To evaluate the impact of the x-ray source arrangements across the different anatomical features (i.e., microcalcifications, masses, and linear fibrils) in the phantom, the stacked phantom was imaged in two configurations: one with the flat region of the semi-circular shape placed posteriorly, and one with the flat region placed anteriorly.

Final qualitative assessment of image quality in anatomically realistic scenarios was obtained by imaging a roughly 5-cm-thick sample of bovine muscle (brisket) having an approximate interstitial fat-to-muscular tissue ratio of about 7% by weight.

3 |. RESULTS

3.1 |. Hole-stimulus-phantom with anatomical clutter

Figure 5 shows DBT reconstructions for a DBT image plane placed at the location of the hole stimuli, as a function of the 2D MXA configuration. For the mammography arrangement (i.e., using only the single central source in the array), shown in Figure 5a, the full overlap of clutter and feature planes hampers the visualization of the hole stimuli, regardless of their diameter and depth. This is seen more clearly in the zoom-in views, for a diameter $D=1.59\text{mm}$ and depth $h=1.6\text{mm}$ hole impulse, placed posteriorly $(y\approx 33\text{mm})$, and another placed anteriorly $(y\approx 100\text{mm})$. Beyond limited visibility of image features from clutter overlap, anterior-most regions of the phantom fell outside of the collimated beam and were therefore truncated. The truncated region is shown as a dark horizontal band in Figure 5a.

FIGURE 5.

Image results for the clutter and stimuli phantom. (a) Mammography view, acquired at the same dose as the DBT acquisitions, showing severe clutter overlap that hampers visualization of image stimuli. (b) DBT planes at the location of the stimuli for 1D MXA DBT and (c–e) for 2D MXA setups with increasing gap between the individual MXAs: (c) $\delta =19\text{mm}$, (d) $\delta =38\text{mm}$, and (e) $\delta =57\text{mm}$. Magnified views for the stimulus with the lowest diameter and thickness at $y~33\text{mm}$ (pink) and $y~100\text{mm}$ (cyan), showing a progressive increase in the conspicuity of small, low-contrast features with increasing $\delta$ for 2D MXA configurations.

The 1D DBT with a single MXA (Figure 5b) resulted in improved visualization of the hole stimuli compared to mammography (Figure 5a) across the system field of view, consistent with previous results on DBT image quality measurements. The magnified views show better conspicuity of the posterior features, compared to the anterior features, where interslice clutter was more noticeable—reducing conspicuity. The 2D MXA configurations yielded more homogeneous sampling across the field-of-view compared to the 1D linear geometry, where sampling density at anterior locations of the FOV was reduced due to the large cone angle defined by each source element in the MXA.

Similar to mammography, the 1D MXA configuration resulted in the truncation of the anterior-most regions of the FOV. In addition to increased sampling density at anterior locations, 2D MXA source arrangements resulted in extended FOV, resulting in increased coverage of anterior-most regions of the phantom.

The visual conspicuity of the anterior stimuli increased with increased separation between the MXAs in the 2D MXA configuration, showing a marked reduction of interslice clutter for $\delta =57\text{mm}$, the geometry that minimized cone angle effects for anterior locations.

Reduced in-signal-plane clutter at anterior locations with increasing array separation $\left(\delta \right)$ can be seen in Figure 5, and these results are shown quantitatively in Figure 6. In agreement with visual evaluation, the 2D MXA configurations reduced the clutter noise across the tomosynthesis plane. The distribution of clutter noise, measured for the ROIs in Figure 6a, is depicted in Figure 6b, which shows a reduction in median (across ROIs) clutter noise for increasing separation $\delta$ between the two source arrays. Consistent with the observations of similar clutter noise at posterior locations for 1D and 2D-MXA configurations, the distributions in Figure 6b shows similar noise levels at the bottom range (lowest bar on the box plots), with a 10-percentile at 0.0047 A.U. for the 1D MXA $(\delta =0)$ and 0.0046 A.U. for the 2D MXA $(\delta =57\text{mm})$. The improvement achieved by 2D-MXA configurations is reflected as a reduction of clutter noise in regions with large baseline noise (largely anterior ROIs), resulting in a reduction of 10.2% in median clutter noise and 16.9% in the 90-percentile. The reduction in clutter noise was statistically significant $(p<5\times {10}^{-4})$ when comparing $\delta =0\text{mm}$ and $\delta =57\text{mm}$.

FIGURE 6.

Estimated anatomical clutter noise as a function of the MXA configuration. (a) Illustration on one of the DBT planes used for measurement of clutter noise. The ROIs used for the estimation are depicted as yellow squares. (b) Distribution of measured clutter noise across the test ROIs, showing overall lower clutter noise (reduced median) and more homogeneous clutter noise across ROIs (reduced interquartile range) for increasing separation between individual MXAs in the 2D MXA configurations. The reduction in anatomical clutter for 2D sampling was statistically significant $\left(p<5\times {10}^{-4}\right)$.

CNR values as a function of the MXA configuration and feature depth are shown in Figure 7. In agreement with previous results in clutter noise and feature conspicuity,²² high-contrast, deep features $(h=4.8\text{mm})$ are better resolved, yielding CNR > 4 for all configurations and hole diameters. For these high contrast features (Figure 7a), the effect of interslice clutter is mitigated with decreasing feature diameter, illustrating the mid-to-low frequency nature of the clutter noise. Furthermore, visual inspection of clutter noise at distal locations for 1D- versus 2D-MXA configurations (Figure 5) indicates a lower frequency distribution of clutter noise with 2D-MXA setups. This visual observation is confirmed by the larger impact in CNR of $\delta$ for small diameter high contrast features (e.g., for $D<4\text{mm},h=4.8\text{mm}$ in Figure 7a). For the most extreme case, where $D=1.59\text{mm}$ and $h=4.8\text{mm}$, the 2D-MXA configuration with $\delta =57\text{mm}$ yielded CNR = 5.9, compared to CNR = 4.6 for the 1D-MXA setup $(\delta =0)$. In agreement with conventional assumptions on DBT image quality, for lower contrast structures (i.e., lower $h$) overall CNR values are lower. For example, for $h=3.2\text{mm}$ (Figure 7b), CNR values remained stable (at CNR ~ 3.7) across the range of feature diameters explored. Consistent with results on high-contrast features, for $h=3.2\text{mm}$, similar trends are observed for 2D-MXA configurations, with a higher impact for very small diameters, yielding an increase in CNR of 0.25 for $\delta =57\text{mm}$ and $D=1.59\text{mm}$.

FIGURE 7.

CNR measurements as a function of diameter of the stimulus, for a stimulus thickness of $h=4.8\text{mm}$ (a), $h=3.2\text{mm}$ (b), and $h=1.6\text{mm}$ (c). Error bars are $\pm 1\sigma$.

In the case of the low-contrast stimuli $(h=1.6\text{mm})$, comparable to some soft-tissue masses, the contrast is of a magnitude similar to the clutter noise, reducing the CNR and reversing the trend observed for features with higher contrast-to-background, as shown in Figure 7c. In this case, small stimuli $(D=1.59\text{mm})$ yield a baseline CNR = 1.2 for the 1D-MXA configuration, compared to CNR = 3.0 for the largest diameter $(D=6.35\text{mm})$. The reduction in clutter noise provided by the 2D-MXA configurations resulted in a monotonic increase in CNR with increasing $D$, with CNR = 1.9 and CNR = 3.4 for $D=1.59\text{mm}$ and 6.35 mm, respectively, for $\delta =57\text{mm}$.

Quantification of the frequency distribution in individual tomosynthesis planes is illustrated in Figure 8 as in-plane spectral power distributions measured on a tomosynthesis plane focused on the clutter region for anterior and posterior regions, denoted $S_B^A$ and $S_B^P$, respectively. Configurations using a 1D MXA scan showed a loss in sampling of vertical frequencies in the image (aligning with the AP axis of the scanner—y axis in Figure 3) for anterior locations (Figure 8c), compared to posterior locations (Figure 8b). The null cone in the frequency space is caused by a noticeable reduction of sampling density associated with the large cone angle (in $y$) of the 1D configuration at anterior locations. Consistent with previous image results, 2D MXA setups resulted in better sampling at anterior locations, particularly at frequencies aligned with the antero-posterior axis of the scanner, due to the drastic reduction of cone angle effects provided by the second array of sources. The improved sampling with the 2D-MXA geometry at $\delta =57\text{mm}$ is demonstrated by the very small differences in the power spectra between the posterior (Figure 8d) and anterior (Figure 8e) locations. By comparison, the anterior 1D MXA results (Figure 8c) show a much larger null cone in the 2D NPS.

FIGURE 8.

Power spectra measured on the background clutter, for posterior and anterior locations. (a) DBT slice and ROI setup used for measurement of background power spectrum using partially overlapped ROIs. The set of ROIs used for computation of posterior regions of the field of view is illustrated in green, and for anterior locations in blue. Background clutter power spectra are shown for the 1D MXA configuration at posterior (b) and anterior (c) locations. Background clutter power spectrum for the 2D MXA configuration with $\delta =57\text{mm}$ are shown for the posterior (d) and anterior (e) regions.

3.2 |. Slice-sensitivity profile

Figure 9 shows slice-sensitivity metrics, for the 1D MXA and the 2D MXA setup with $\delta =57\text{mm}$. DBT views of the thin brass feature along the system $z$ plane show apparent lower interslice blur for the 2D-MXA configuration, in line with reduced clutter noise observed above. SSPs were obtained along the dashed lines indicated in Figure 9a,b and are displayed in Figure 9c for the nominal reconstruction protocol. FWHM values were computed for both profiles at the location of the dashed black line in Figure 9c, yielding FWHM = 5.95 mm for the 1D MXA, compared to FWHM = 5.34 mm for 2D MXA with $\delta =57\text{mm}$.

FIGURE 9.

Slice sensitivity profile measured on the planar brass impulse feature. Axial view of the feature placed at a central location in the system field of view, for the 1D MXA configuration (a) and for the 2D MXA configuration with $\delta =57\text{mm}$ (b). (c) Slice sensitivity profiles taken at the central location of the feature indicated with the dashed line in (a, b). The horizontal dashed line shows the half-maximum position used to measure profile width, pointing to a net benefit in FWHM from the 2D MXA configuration. (d) FWHM obtained for the 1D and 2D MXA configurations as a function of the cutoff frequency of the interslice filter. Error bars show the FWHM standard deviation across slices of the brass feature.

The superior SSP of the 2D-MXA geometry is evident in Figure 9d, which illustrates the FWHM of the SSP as a function of $\beta$. FWHM values for the 1D MXA remained larger across the explored range of $\beta$, indicating a net benefit in terms of thinner tomosynthesis slice thickness with the 2D-MXA geometry, with more noticeable differences for lower values of $\beta$. The observed differences between 1D MXA and 2D MXA were statistically significant, with $p$ values ranging between $p=0.008$ for $\beta =0.1$ to $p=0.024$ for $\beta =0.9$.

3.3 |. Anatomically realistic breast phantom

Images of the commercial breast phantom are illustrated in Figure 10, for a conventional imaging setup with the flat region of the semicircular phantom placed at the posterior region (i.e., close to the chest wall). Consistent with the hole stimulus phantom results, the mammography view of the commercial breast phantom exhibited very little conspicuity of high contrast microcalcification targets at the posterior region. Similarly, soft-tissue masses were not discernible from anatomical clutter in this region, with the exception of the largest mass. DBT reconstructions exhibited greater conspicuity and reduced background anatomical clutter. Similar image quality was observed across 1D and 2D-MXA configurations as shown in Figure 11.

FIGURE 10.

(a) Mammography view of the anatomical test phantom. (b–e) DBT planes at the central slab of the phantom illustrate the visibility of the different image features as a function of the MXA configuration, for (b) 1D MXA and (c–e) 2D MXAs with increasing gap between the arrays: (c) $\delta =19\text{mm}$, (d) $\delta =38\text{mm}$, and (e) $\delta =57\text{mm}$.

FIGURE 11.

Zoom-in details of high-contrast CaCO₃ features with fine (0.196 mm) and large (0.290 mm) diameter, and of the smallest low-contrast mass sphere (1.8 mm diameter), for mammography (a, f, k), single MXA (b, g, l), and 2D MXA configurations with $\delta =19\text{mm}$ (c, h, m), $\delta =38\text{mm}$ (d, i, n), and $\delta =57\text{mm}$ (e, j, o). For DBT configurations, the left column illustrates the visibility of the feature placed at a posterior location with respect to the reference MXA position, and the right column shows the same feature when imaged at anterior locations. Blue arrows in $\left(l\right)$, (m), (n), and (o) point to the location of the spherical mass.

Figure 11 shows zoomed-in details of the CaCO₃ microcalcification surrogates for two feature sizes (0.196 and 0.290 mm) and for the smallest soft-tissue mass (1.8 mm). For DBT configurations, the left column illustrates the visibility of the feature placed at a proximal location relative to the reference MXA position (i.e., close to posterior regions of the field-of-view), and the right column shows the same feature when imaged at distal locations (i.e., close to anterior regions of the field-of-view). The breast phantom included structured anatomical clutter of higher frequency than the spherical clutter in Section 3.1., which resulted in a different appearance of clutter noise. This high-frequency clutter noise presented itself as rippling artifacts, particularly conspicuous in anterior regions of the field-of-view (see, e.g., Figure 11b). The zoomed-in views show increased conspicuity of microcalcification features with 2D-MXA configurations when imaged at distal locations. This effect is particularly noticeable for the 0.290 mm CaCO₃ features, in which the reduction of anatomical clutter yielded reduced high frequency rippling artifacts, resulting in better contrast between the CaCO₃ features within the surrounding artifacts.

Similar observations relate to the visibility of the soft-tissue mass, illustrated by blue arrows in Figure 11, showing a reduction of anatomical clutter and increased conspicuity of the soft-tissue mass in distal locations with increasing $\delta$, while preserving its conspicuity at proximal locations.

Finally, the effects of 1D and 2D MXA sampling were verified qualitatively in images of the bovine muscle and interstitial adipose, as illustrated in Figure 12. All DBT configurations exhibited improved visibility of muscular and interstitial tissues (Figure 12) compared to the single mammographic view. The zoomed-in views show improved conspicuity of two naturally occurring small adipose inclusions with increasing $\delta$, resulting from a reduction of interslice clutter, consistent with observations in the phantom experiments.

FIGURE 12.

Image results on samples of muscle and interstitial adipose tissue. (a) Mammography image, (b) DBT with a 1D MXA, and (c–e) DBT with 2D MXA at varying source separation distance. (f–j) Zoom-in views of interstitial fat features for all configurations.

4 |. DISCUSSION

The potential of a 2D array of grid-controlled thermionic cathodes integrated into an enclosed vacuum environment with a long rotating shaft of anodes has been demonstrated for DBT. Building from decades of conceptual understanding on the superiority of a 2D source distribution over a 1D source distribution, the quantitative experiments shown here demonstrated the advantages of the 2D source configuration in this study. Experimentally, three 2D source distributions were studied, with the spacing between two, 11-source linear arrays varied from $\delta =0$ (a single linear array) to 19, 38, and 57 mm. Not surprisingly, the widest separation of $\delta =57\text{mm}$ demonstrated the best overall performance.

The effects of source geometry on anatomical clutter were illustrated and quantified in Figures 5–6 for mammography, a 1D source array, and three 2D source array configurations. The magnitude of anatomical clutter noise in the focal plane was reduced monotonically over this spectrum of source geometries and was minimized with the 2D MXA configuration with greatest source separation. The corresponding effect on contrast-detail performance and CNR (at fixed total exposure) was quantified in Figure 7. CNR was seen to increase with source separation $\left(\delta \right)$ as expected. An apparent dependence of CNR on stimulus diameter was attributed to non-stationarity in the 3D point spread function, the geometry of the phantom (namely, that the stimuli were flat-bottomed cylinders, not spheres), and the limited nature of CNR as a performance metric (vs., for example, contrast- and frequency-dependent SNR). For low-contrast stimuli, CNR increased with stimulus diameter, whereas CNR decreased with stimulus size for high-contrast stimuli. These experimental limitations likely benefited the apparent performance of the mammography system, for which x-rays are more normal to the surface of the detector and aligned roughly parallel to the cylindrical stimuli.

The reduction in background clutter noise and power spectrum with 2D MXA geometry was further demonstrated in Figure 8. For the posterior region of the clutter phantom—near the chest wall side of the field of view—the performance of the 1D MXA was similar to 2D MXA, likely because the $\delta =0$ array position was essentially directly above the posterior region, reducing the cone angle. For the anterior position, however, the 1D MXA carries a large cone angle in this direction, whereas the 2D MXA with $\delta =57\text{mm}$ is located directly over the anterior region, giving a small cone angle and improved frequency sampling. The Fourier domain “null cone” exhibited a region of unsampled frequencies approximately equal to the cone angle of the x-ray paths, as expected.

The experimental studies summarized above were designed to assess the impact of 1D and 2D sampling on out-of-plane clutter at the tomosynthesis plane (i.e., clutter noise). To this end, the experiments posed an equidose scenario, in which 1D MXA arrangements involved acquiring 11 source positions with twice the x-ray exposure compared to each of the 22 views in 2D MXA configurations. In this case the sampling density obtained at regions proximal to the chest wall is equal in all configurations using either 1D or 2D MXA, albeit with larger per-view quantum noise in 2D MXA setups. This increase in quantum noise is partially captured in the clutter noise metrics. For example, the minimum clutter noise value (obtained for regions proximal to the chest wall) in the distributions in Figure 6b was larger for 2D MXA. This increase is partially attributable to higher quantum noise in this region with the 2D MXA.

Alternative designs for 1D MXA configurations are possible. For example, a configuration using a denser 1D MXA with 22 sources distributed over the same distance of 23 cm would result in similar equidose conditions but denser sampling at chest wall locations. Preliminary studies on task-based DBT sampling optimization pointed to modest gains in performance with denser arrays for high-frequency imaging tasks (e.g., microcalcifications) with, however, minimal impact on clutter noise.²⁶

The SSP (Figure 9) was similarly improved with the 2D MXA compared to the 1D MXA. Depending on the choice of slice thickness filter (Figure 9d), the SSP for the 2D MXA was substantially improved over the 1D MXA, and for a lower normalized frequency cutoff $(\beta =0.1)$, the improvement was even more pronounced.

Images of a commercial breast phantom (Figures 10–11) demonstrated that larger “microcalcifications” (0.290 mm) were hardly seen in the mammography and were more conspicuous in both 1D MXA and 2D MXA—both anteriorly and posteriorly. Meanwhile, masses were far more conspicuous in the 2D MXA $(\delta =57\text{mm})$ than in the 1D system. A similar increase in lesion conspicuity was seen for fat inclusions in a bovine tissue phantom.

The results presented in this work offer a preliminary assessment of image quality with 2D DBT sampling in physical experiments using a discrete set of 2D MXA and a prototype 1D MXA design. The prototype MXA used in this study was designed to perform optimally for DBT with 1D sampling and with geometrical configurations akin to those used in current state-of-the-art DBT systems. Therefore, certain design choices might not be optimal for other breast imaging modalities or protocols. For example, the prototype 1D MXA featured a single focal spot configuration per source element, with a fixed nominal size of 0.3 mm. This focal spot size results in minimal focal spot blurring for screening mammography and for DBT protocols used in screening applications. Despite the suitability of the current MXA focal spot size for screening protocols, certain diagnostic procedures require a smaller focal spot size. For example, magnification mammography, which imparts geometrical magnification for the acquisition of high-resolution mammographic views of ROIs within the breast, requires a focal spot size of ~0.1 mm.²⁷ It is worth noting that the reported focal spot size is a design choice of the current MXA. Alternative designs resulting in a smaller focal spot could be achieved with a modest evolution of the MXA concept. For example, one could envision a design with a modified focusing cup, such as one incorporating a second grid electrode on the central element of the MXA, to produce a 0.1 mm focal spot. This would enable the acquisition of magnification mammography views in selected, single-source protocols.

We note that when using the 1D MXA to generate a 2D spatial distribution of sources, different heel-effect properties are produced when compared to the proposed shared-disk 2D MXA. The experimental arrangement used in this work emulated the 2D MXA by shifting a single 1D MXA with the result that anterior locations of the FOV consistently received a lower x-ray flux than posterior locations, yielding higher quantum noise in the reconstructed volume at anterior regions of the FOV. In contrast, the shared anode concept results in anterior and posterior positioned arrays producing reversed heel directions, that is, the x-ray intensity profile due to the heel effect will drop towards anterior regions of the FOV for the array of sources closer to the chest wall, and it will drop at posterior regions for sources located further from the chest wall. Similarly, regions of the FOV with lower attenuation from the heel effect result in smaller effective focal spot and increased spatial resolution. In the proposed 2D MXA, small-focal-spot regions will be located at posterior regions of the FOV for the anterior source array and at anterior regions for the posterior source array.

The complementary nature of the heel profiles in the shared anode 2D MXA will help balance the overall x-ray intensity and spatial resolution throughout the FOV during the joint reconstruction process. In comparison to the experimental setup used in this work, the resulting 2D MXA configuration will lead to higher noise levels at the chest wall, increasing the minimum noise values as $\delta$ increases in the distribution shown in Figure 6. At the same time, it will result in lower noise levels in the anterior regions, thereby reducing the upper limit of the distribution for $\delta >0$. It is worth noting that the somewhat simple weighting approach used in the joint reconstruction method to combine the contribution of multiple measurements to a single voxel (Section 2.2) might not be optimal to fully exploit the complementary noise-resolution properties of the 2D MXA. Ongoing work aims at the development of model-based joint reconstruction approaches with spatially varying regularization penalties, that have shown potential for equalization of noise and spatial resolution throughout the FOV in scenarios with spatially varying sampling density²⁸ and noise properties.²⁹

The experiments on the impact of 2D sampling on DBT showed improved image quality with increasing separation of the linear arrays $\left(\delta \right)$, for the evaluated imaging tasks, including: (i) soft-tissue (low contrast), mid-frequency tasks such as detection of soft-tissue mass lesions (Figures 10–11); and (ii) high-contrast, high-frequency tasks pertinent to detection of microcalcifications. However, the results were limited by the experimental setup: the maximum achievable $\delta$, and the number of configurations (values of $\delta$) that could be investigated with the current setup. A more complete picture for optimization of the 2D MXA under development falls to ongoing work, including analytical models of task-based detectability²³ accounting for the effects of cone angle on spatial resolution and noise power spectrum in tilted beams.³⁰

Scan time is a similarly important design criterion for DBT systems and stands to benefit substantially from 1D or 2D MXA geometries. Targeting a scan time less than 1 s, the required mAs for a similar x-ray spectrum across five clinical Hologic DBT systems at UC Davis, assuming 15 pulses per scan, was computed for various phantom thicknesses to be: 2 cm thick—38 mAs, 4 cm thick—51 mAs, 6 cm thick—69 mAs, and 8 cm thick—87 mAs. For the MXA system, operating at 115 mA with 35 ms pulses for 22 sources delivered a total of 88 mAs in about 1.0 sec, illustrating the potential of MXA to deliver sub-second DBT.

5 |. CONCLUSIONS

The results obtained in this investigation using physical experiments demonstrated the advantages of a 2D source geometry for DBT compared to the conventional 1D line of x-ray sources, due to both improved sampling in the frequency domain and reduced SSP. Based on the observations presented in this work, ongoing research aims at comprehensive optimization of geometrical parameters via task-based detectability metrics as well as physical implementation of a 2D MXA prototype system. The improvement in sampling and image quality from the 2D source geometry will come with a combined advantage of a major reduction in scan time. Considering these observations, we conclude that 2D MXA thermionic x-ray sources have the potential to substantially improve DBT imaging in comparison to a conventional 1D scan.